In the large \(N\)  limit it is well known that the index
is reproduced on the AdS side as the supergravity index
\({\cal I}_{\rm KK}=\mathop {\rm Pexp}\nolimits i_{\rm KK}\)  [1]}.
\(\mathop {\rm Pexp}\nolimits \)  is the plethystic exponential defined in the next section.
\(i_{\rm KK}\)  is the single-particle index
\(i_{\rm KK}&=\frac{qu_1}{1-qu_1}+\frac{qu_2}{1-qu_2}+\frac{qu_3}{1-qu_3}-\frac{q^{\frac{3}{2}}y}{1-q^{\frac{3}{2}}y}-\frac{q^{\frac{3}{2}}y^{-1}}{1-q^{\frac{3}{2}}y^{-1}}.\) 
